# The multiplier effect The “multiplier” tells us the change in

advertisement ```The multiplier effect
The “multiplier” tells us the change in equilibrium GDP from a given change in desired spending,
say a change in investment brought about by either a shift of the investment demand curve or a change in
∆Ye
real interest rates. Mathematically, we desire a formula for
where Ye is equilibrium GDP.
∆Ig
Suppose GDP is given by the formula Y = C + Ig where C takes the linear formula C = a + bY. As
in math notes 27.1, “Consumption and saving,” a is autonomous consumption spending and b is the MPC.
 1 
Solving for equilibrium GDP, we obtain the general result Ye = 
 x (a + Ig).
1− b 
If Ig changes by some amount ∆Ig, then Ye will change by some amount ∆Ye:
 1 
 1 
 1 
Ye + ∆ Y e = 
 x (a + Ig + ∆Ig) = 
 x (a + Ig) + 
 x ∆Ig. We can simplify this by
1
−
b
1
−
b




1− b 
 1 
 1 
subtracting Ye and its equivalent 
 x (a + Ig) from both sides to obtain ∆Ye = 
 x ∆Ig. Now
1
b
−


1− b 
∆Ye  1 
divide ∆Ig on both sides to obtain our desired result:
=
.
∆Ig  1 − b 
∆Ye
=
Since b is the MPC and (1 – b) is the MPS, this can be expressed alternatively as
∆Ig
1

  1 
 1   1 

 =
 . For example, if the MPC is .75, the multiplier is 
 =
 = 4, so that
 1 − MPC   MPS 
 1 − .75   .25 
if ∆Ig = \$5 billion, ∆Ye = \$20 billion.
 1 
Incidentally, the term 
 has an alternate interpretation. Suppose we have an infinite sum of
1− b 
the form 1 + b + b2 + b3 + b4 + …. We don’t know what, if anything, this sum is equal to, but suppose it
converges to some value Z. That is, Z = 1 + b + b2 + b3 + b4 + …. If we multiply each side of this
equation by b, we would have bZ = b + b2 + b3 + b4 + b5 + …. Next, subtract bZ from Z: Z – bZ = 1, as
every term except the first term of Z cancels out. We can factor out Z from the term on the left and as
 1 
long as b does not equal 1, divide both sides by 1 – b to obtain Z = 
 . That is, if the infinite sum
1 − b 
 1 
converges at all, it will converge to 
 . Note that for b &gt; 1 the sum is infinite. However, if b &lt; 1 it
1− b 
 1 
can be shown that the sum converges to the value 
 which you will recognize as the multiplier.
1 − b 
This has an economic interpretation: the change in equilibrium GDP from a given change in investment
can be seen as the sum of successive “rounds” of additional spending, with the amount at each round a
constant proportion (equal to the MPC) of the prior round.
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